Fast Lyapunov Indicator

THE FINE STRUCTURE OF HAMILTONIAN SYSTEMS REVEALED USING THE FAST LYAPUNOV INDICATOR... 

Different methods have been developed either for a rapid computation of the LCIs (Cincotta and Simo 2000) or for detecting the structure of the phase space (chaotic zones, weak chaos, regular resonant motion, invariant tori). Especially for this last purpose we quote the frequency map analysis (Laskar 1990, Laskar et al. 1992, Laskar 1993, Lega and Froeschle 1996), the sup-map method (Laskar 1994, Froeschle and Lega 1996), and more recently the fast Lyapunov indicator (hereafter FLI, Froeschle et al. 1997, Froeschle et al. 2000) and the Relative Lyapunov Indicator (Sandor et al. 2000). The definitions and comparisons between different methods including a preliminary version of the FLI have been discussed in Froeschle and Lega (1998, 1999). 

The paper is organized as follows in Section 2 and 3 we define the Fast Lyapunov Indicator and give some examples on the 2 dimensional standard map and on a Hamiltonian model. The special case of periodic orbits will be detailed in 4 and thanks to a model of linear elliptic rotation we will be able to recover the structure of the phase space in the vicinity of a noble torus. The use of the FLI for detecting the transition between the stable Nekhoroshev regime to the diffusive Chirikov s one will be recalled in Section 5. In 6 and 7 we will make use of the FLI results for the detection of the Arnold s diffusion. 

Figure 2. Variation of the Fast Lyapunov Indicator with time, a) Orbits of the standard map of equation (5) with e = 0.7. The upper curve is for a chaotic orbit with initial conditions x(O) = 10 4,j/(0) = 0, the second one is for a weak chaotic orbit with x(0) = 0.4839, y(0) = 0, the third one is for a non resonant orbit with x(0) = 2.5,2/(0) = 0 and the lowest one is for a resonant orbit with x(0) = 1.2, y(0) = 0. b) Orbits of Hamiltonian (6) with e = 0.004. The upper curve is for a chaotic orbit with initial conditions -Zi(O) = 0.2849, the second one is for a weak chaotic orbit with /1(0) = 0.2309, the third one is for a non resonant orbit with /i(0) = 0.2204 and the lowest one is for a resonant orbit with /1(0) = 0.342. The other initial conditions are I2(0) = 0.16, M0) = 1, >i(0) =

Froeschle, C., Gonczi, R. and Lega, E. (1997). The fast Lyapunov indicator a simple tool to detect weak chaos. Application to the structure of the main asteroidal belt. 

Lega, E. and Froeschle, C. (1997). Fast Lyapunov Indicators. Comparison with other chaos indicators. Application to two and four dimensional maps, in The Dynamical Behaviour of our planetary system., Kluwer academ. publ., J. Henrard and R.Dvorak eds. 

Lega, E. and Froeschle, C. (2001). On the relationship between fast Lyapunov indicator and periodic orbits for symplectic mappings. Celest. Mech. and Dynamical Astronomy, 15 1-19. 

The transition of spectra from structured to unstructured ones is not described by a theorem, but has been studied numerically in Guzzo, Lega and Froeschle (2002) by comparing the geometry of resonances of a given system computed with the Fast Lyapunov Indicator with the structure of the spectra of an observable computed on well selected chaotic solutions. More precisely, in Froeschle et al. (2000) we estimated with the FLI method that the transition between Littlewood and Chirikov regime for the Hamiltonian system ... 

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